Question

Newton's Law of Cooling says that the rate at which a body cools is proportional to the difference C in temperature between the body and the environment around it. The temperature f(t) of the body at time t in hours after being introduced into an environment having consistent temperature T_0 is f(t)=T_0+C e^-kt, where C and k are constants.A cup of coffee with temperature 72°C is placed in a room with temperature 12°C. After 1 hour, the temperature of the coffee is 42°C. Complete parts (a) through (c).(a) Write an equation to model the data.(b) Estimate the temperature after 25 minutes.(c) How long will it take for the coffee to cool to 35°C?

Answer 1

To find the equation of the temperature of the coffee, we must use the information given to find the missing constants

$f(t)=T_0+C\cdot e^{-kt}$

To find C we can use that

$t=0;f(t)=72$

remembering that T0= 12°C

Replace in the equation

$\begin{gathered} 72=12+C\cdot e^{-(k\cdot0)} \\ 72=12+C\cdot1 \\ C=72-12 \\ C=60 \end{gathered}$

Now to find K we use

$t=1;f(t)=42$

replace data in the equation and find k

$\begin{gathered} 42=12+60\cdot e^{-k(1)} \\ 42-12=60e^{-k} \\ 30=60e^{-k} \\ \frac{30}{60}=e^{-k} \\ \frac{1}{2}=e^{-k} \\ \ln (\frac{1}{2})=-k \\ -\ln (\frac{1}{2})=k \\ k\cong0.693 \end{gathered}$

The equation is

$f(t)=12+60e^{-0.693t}$